For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation.pdf. For this Number Talk, I am encouraging students to represent their thinking using an array model.

Task 1: 18/3

For the first task, 18/3, some students drew an array, placed 18 on the inside, 3 on one side, and a ? mark on the unknown side: 18:3 = 6. Others decomposed the 18: 15:3 + 3:3 = 6.

Task 2:36/3

During the next task, we discussed how to represent 36/3. Many students referred to the first task and pointed out that the dividend doubled so the quotient will double: 36:3 = 2(18:3).

Task 3:300/3

During the next task, we quickly discussed how this division problem could be represented and didn't waste any time before moving on to the next task!

Task 4:636/3

As the tasks increased with difficulty, so did student engagement. Here, a student decomposes 636 into 600 + 30 + 6 to solve this equation: 636:3.

Task 5:1272/3

For the final task, student's were at first intimidated by a dividend in the thousands. With time, most students realized how they could use the previous tasks to solve this task: 1272:3.

Resources

To begin today's lesson, I introduced the goal: I can find the factors for numbers up to 100. I explained: Over the past few lessons, you have learned how to identify the factors for numbers using the prime factorization method and the u-turn method. Today you get to use your factor-finding skills to play a game!

Reviewing Multiples & Factors

To review Multiples and Factors. I asked students to turn and talk: Explain the difference between factors and multiples!

Reviewing Prime & Composite Numbers

Next, I referred to the I'm Prime Chant and began singing, "I'm Prime... P-R-I-M-E... the only factors are one and me. I then asked students to turn and talk: How many factors do all prime numbers have? (Two... One and itself.)

Player A chooses a number on the game board and circles it. This will be Partner A’s score for that round.

Using a different color, Partner B circles all the proper factors of Player A’s number.

The proper factors of a number are all the factors of that number except the number itself. Partner B lists the factors. The sum of those factors is Partner B’s score for that round.

Player B then circles a new number. Player A circles all the remaining factors of that number. Then, play continues in this manner.

The players take turns choosing numbers and circling factors.

If a player circles a number that has no factors left which have not been circled, then that player does not get points for the number circles and loses his or her turn.

The game ends when there are no more numbers left with uncircled factors.

The player with the larger sum of factors and products is the winner.

Modeling the Factor Game

I decided to begin by teaching students Version A first as counting points for prime numbers seemed easier than adding the sum of all the factors. I also began by covering up half of the Factor Game to 30 game board to build a staircase of complexity: working with factors up to 15, then up to 30, and then up to 100. To make sure students were successful playing the game, I chose a student volunteer as my partner and modeled the directions step-by-step. Instead of circling with two different crayon colors, we used multi-colored dry erase markers.

Resources

I passed out materials (a page protector with the two game boards (Factor Game to 30.and Factor Game to 100) front to back and multi-colored dry eraser markers) to each pair of students. Choosing partners is always easy in my classroom as students are already strategically placed in desk groups based upon behaviors and ability levels.

Playing the Factor Game up to 15

Students couldn't wait to begin playing! At first, I asked them to use half of the Factor Game to 30, just as I had modeled. During this time I conferenced with each group and asked questions to check for understanding and to push student thinking:

What strategies are you using?

What have you noticed about this game?

What is your goal right now?

How do you know if a number is prime or composite?

Which numbers are the best to choose?

Which numbers are the worst to choose?

With each of these questions, I would also ask: Can you explain why? This simple followup question really pushes students to construct viable arguments based on evidence (Math Practice 3).

As students finished the first game, they eagerly came up to be and asked if they could play the full Factor Game to 30 board. Before letting some students move on, I Conferenced with Students about Strategies. I loved how one student pointed out that the game board to 30 might have an odd number of prime numbers instead of an even number of prime numbers!

Students then moved on to playing with the full game board: Students Playing Game up to 30. After most students had played the Factor Game up to 15 and up to 30, I knew we were ready for a change!

Students had become quite comfortable with Version A. of the game.... so comfortable that they began just picking all the prime numbers and ignoring the composite numbers as they were literally "pointless!" I realized that Version A encouraged students to focus mostly on prime numbers instead of identifying factors of numbers under 100. It was at this point that I wanted to raise the standard by teaching students Version B of this game!

I called students up to the front carpet and again, chose a volunteer to help me model the new directions. It didn't take very long to model the game as students were already familiar with the game. Using the Factor Game to 30, I explained: We are going to make this game a little harder! For this version of the game, you'll get 2 points if you circle the 2 and you'll get 15 points if you circle the 15 and you'll get 30 points if you circle 30. Let me show you how this works. (I explained while circling a projected game board.) Let's say partner A circles the 30. Partner A will get 30 points. Some students gasped at the number of points! However, Partner B will get to circle all the factors for 30. What are the factors for 30? (Students helped by identifying: 1, 2, 15, 3, 10, 5, 6. Who made more points... Partner A or B? Students added up the points and were surprised to find that Partner B actually had 42 points!

As students began playing, it was immediately clear that this game required more mathematical thinking than Version A and would also better today's goal. I so was glad we made the switch! I proudly watched as students were looking for and making use of structure within this game (Math Practice 7).

Playing the Factor Game up to 100

Again, students were eager to move on to the final level of today's activities. I loved watching students apply their factor-game-playing-skills that were developed with the easier game boards! I continued to conference with each group during this time: Students Playing Game Board up to 100.